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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA subset $A \\subseteq X$ is an *$F_\\sigma$ set* if there are [[closed sets|0ZCK8DQYN1]] $F_1, F_2, F_3, \\dots$ of $X$ with\n$$A = \\bigcup_{n=1}^{\\infty} F_n.$$\n\n## Remarks\n\nA set is $F_\\sigma$ if and only if its complement is a [[G-delta set|V1TBNP8R3S]], by De Morgan's laws.\n\nExample: $\\Q = \\bigcup_{q \\in \\Q} \\set{q}$ in $\\R$.\nEach $\\set{q}$ is closed and $\\Q$ is [[countable|01TQB88H4C]], so $\\Q$ is $F_\\sigma$.\n\nExample: $(0, 1) = \\bigcup_{n \\ge 3} [1/n, 1 - 1/n]$, so this open interval is $F_\\sigma$.\nIn a [[metric space|0VKD1TP7F3]], every open set is $F_\\sigma$.\n\nNon-example: the set $\\R \\setminus \\Q$ of irrational numbers is not $F_\\sigma$ in $\\R$.\nOtherwise its complement $\\Q$ would be a $G_\\delta$ set, and it is not.\n",
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